寸法効果を考慮したコンクリートの応力度 : ひずみ度曲線の表示式とRC梁のモーメント : 曲率関係への適応

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Concrete Beams

Sachio KOIKE

，

Kenji OHISHI

，

Kazuo OKUFUJI and Nobuyuki OKUYA

寸法効果を考慮したコンクリートの応力度一ひずみ度曲線

の表示式と

RC

梁のモーメントー曲率関係への適用

小池狭千朗・大石健次・奥藤一夫・奥谷伸幸

A

series of experiments are carried out to examine the e妊ectof such parameters as size e百ecton compressive strength and stress-strain curves of concrete， and size e百ecton the moment-curvature relationships of reinforced concrete beams_ Based on the experi -mental data， the stress-strain curves of concrete made with di妊erentsize of specimens are proposed and its application for moment-curvature relationship of reinforced con -crete beams is discussed. The plastic deformational capacity of reinforced concrete beam is remarkably a妊ectedby the size of beam specimen and size of aggregate.

1.INTRODUCTION The stress-strain curves in the stress ascending and descending portion of concrete are markedly a妊ectedby the size of concrete specimen and diam -eter of aggregate. The moment-curvature relation -ships in plastic deformation range of reinforced concrete beams under uniform bending moment are substantially affected by the stress-strain curves of concrete in the compressive zone of beams. Study on inelastic deformational behavior of model con -crete is particularly important to discuss the moment-curvature relationship in plastic deforma -tion range of model reinforced concrete beams with di任erentsize of specimen. Tests and theories on size e妊ectof compressive strength of concrete have been examined by many researchers while no gen -erally accepted numerical expression of model concrete with di妊erentsize. The objectives of the present study are to examine the inelastic deformational behavior and the numerical expression of the inelastic stress -strain relationship of model concrete prisms and to examine experimentally the plastic rotational capacity of reinforced concrete simple beams made wi出 di妊erentsize of specimens and to compare廿le moment-curvature relationships obtained by the experiment and the analysis with the numerical stress-strain relations of model concrete. The authors have been carried out the concrete prism compressive tests and reinforced concrete simple beam bending tests and examined the inelastic deformational behavior of model concrete prisms with di妊erentsize of specimen and di旺erentsize of aggregate up to a large compressive strain and moment-curvature relationship of model beams under fiexural load. 2. EXPERIMENTAL PROCEDURES 2. 1 OUTLINE OF EXPERIMENT The outline of concrete compressive test is shown in Table 1.The variables in the concrete compressive test were the water-cement ratio (W

I

C=0.45， 0.60， and 0.70)， lateral size of specimen (S= D=4.46， 5.55， 7.25， 9.68， 12.48， and 15.00 cm)， and maximum diameter of aggregate (φ= 10， 15， 20， 25， and 30 mm). Six kinds of prism compressive speci -mens having square section with di妊erentsizes were made. The sizes of section (DxD) were 4.46x

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220 Sachio KOIKE， Kenji OHISHI， Kazuo OKUFUJI and Nobuyuki OKUYA

Table 1 Outline of expeliment

Notation ￨Sizeofspecim Water“ Sizeof￨Number

of size of cement aggregate

I

specimen Notation of specimen

sp巴~i~;~

I

~idt~

I

~eight

ratio

D (: S) (cm)

I

H (cm) W/C (%) φ(mm) 盟-PR一♀-J_一盛

E

PR-4 4.46X 4.46 13.48 45 15， 25 PR-5 5.55X 5.55 16.65 PR-7 7.25X 7.25 21.75 60 20 t3 PR-9 9.68X 9.68 29.04 25， 30 t4 PR-12 12.48 x 12 . 48 37.74 t5 PR-15 15.00 x 15.00 45.00 70 15， 25 t1 : Maximum size of aggregate t2 : Size of specimen t3: Uniaxial compressive test t4 : Prism specimen t5 : Water cement ratio

Table 2 Mix proportion of concrete mrartaito Er-cgylent SsPerel.cEiS men S_gi_rz_ae_v eofl (%) (mm) Water 10Ag.series 10-5 191 15Ag.series 15-5 186 60 20Ag.series 20-5 182 25Ag.series 25-5 178 30Ag.series 30・5 174 4.46， 5.55 x 5.55， 7.25 X 7.25， 9.68 X 9.68， 12.48 X 12.48， and 15.00x15.00 cm having height to lateral dimen -sion (h/D) ratio of 3.0. Five kinds of concrete mix proportions having each di妊erentsize of maximum diameter of aggregate were used in the concrete compressive test

2.2 FABRICATION AND CURING OF SPECI-MENS

Ordinary portland cement， Tenryu river sand (maximum size : 5 mm) and Tenryu river gravel were used for concrete. Water-cem巴ntratio (w/c)

of concret巴were45， 60， and 70% by weight.Five

kinds of size of maximum diameter of aggregate (sieve dimension : 10， 15， 20， 25， and 30 mm) were prepared for each concrete series. Mix proportion of concrete is shown in Table 2. The slump of concretes was designed to be 15 cm for all batches. Prism spesimens were cast in steel models in lat -eral position. Each concrete specimens were fa -bricated car巴fully，so as to place the aggregate as inclusion in concrete molds with equal density. Specimens were stored in a labolatory (20土20C， relative humidity 60土5%)until testing. The tests Unit weight(kg/m3) (SrSca/enAnd t)a(p%gEe -) Slump Coment SandI Gravel (cm) 319 737 1092 41 15.0 310 756 1094 42 15.0 304 738 1133 40 15.0 297 723 1162 39 15目。 290 706 1196 38 15.0 of concrete were carried out at the age of about six weeks

2.3 METHODS OF LOADING .AND MEASUREMENT

All the specimens were loaded under the con -stant strain rate of about O.l%/min. by a sti任 compressive testing machine. Longitudinal strains were measured by two displacement transducers attached to the specimen having the gage length of 2D. The complete stress-strain curves were record -ed by an X-Y recorder up to its specified strain(e=

1.2%).

3. TEST RESULTS AND DISCUSSION

3. 1 SIZE EFFECT ON COMPRESSIVE STRENGTH

Fig.1 shows the relationship between com・

pressive strength and size of specimens for the concrete of W /C=60%. Compressive strength of prism specimens increases as the size of specimens becomes larger， but as may be seen from figure，社le curve has a peak at S=9.68cm and there after no increase in the strength is observed and decrease

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Relationships of Reinforced Concrete Beams prFc 3日ot(Kg/cm'l 6目、-PR-C 2S由

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₂ ₄ _s _a ₁₀ STRAIN (x 1 0-3) Fig. 2 Effect of W /C ratio on stress -strain curves with increase in size of specimens S in the range where S is9，68-15.0cm. On the other hand， the compressive strength decreases in parallel with increase in size of aggregate in concrete. These tendencies were already confirmed by the auther's previous study (1).

3. 2 SIZE EFFECT ON STRESS-STRAIN CURVE Fig.2 shows the e妊ect of the water-cement ratio on stress-strain curves. It is obvious from the " 4.00 E 岨 . ... 300 C酒 SO-P円-C .i1 Omm 一温 n H -n u -n u n u 司 4 1 ・

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2 ~ 6 STRAIN {x 日10・SJ 10 Fig. 3 Effect of size of specimen on stress-strain curves figure that the slope of the stress-strain curves in the stress descending portions are larger when the water-cement ratio becomes smaller and compres帯

sive strength becomes larger. Fig.3 shows the e妊ectof the size of specimen with di妊erentsize of aggregate on stress-strain curves. The shapes of stress-strain curves are markedly affected by the siz巴ofspecimens. When the compressive strength becomes larger， the descending portion of the stress-strain curve becomes steeper with the increase in the size of specimen. Furthermore， the ductility of stress -strain curves in the descending portion decreased with the increase of the size of specimen， compared with the specimen series having equal strength level.

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222 Sachio KOIKE， Kenji OHISHI， Kazuo OKUFUJI and Nobuyuki OKUYA Table 3 Expression for stress凶straincurves N ormalized stress-strain curves (5"-E curves) ( 1 ) 5tress ascending portion 5"_-=₁~T _¥_T_._._.Na.E _{1 I}_L_i_'_N_a_{・・・・・・・・・・}₍₁₎ Na-1+EN (11)5tress decending portion q L X

一

d

m

一日一 N

- E

Where: X=B(E-l)m+1， E記 B= 0.023φ

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5+0.199φ+0.04315+0.0109 0.21φ+8.525x10→5+0.805 010975 -2.0 x 10-3φ +2.854 0.21φ+8.525 x 10-35+0.805 Nd= 吉=σ/lio，E=ε/εo Na=l十a(の/100)b a=0.57 ，b=1.0 百=σ/σ

，

。

E=ε/ε

。

B， N d : Empirical constant m: Empirical constant m=0.8 ) 内べ u ( • • • • • • ・ -・・(4) (81 1.0 I~O.B ~O. 6 w 包= トー0.4. m 0.2 E O s C R D i 5 1 ω ( F ﹄

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q J W d 内₄ 0 し z ・1 a m r A n u N a a z v -R s巴ries)

4. NUMERICAL EXPRESSION FOR STRESS-STRAIN CURVE The equation for the stress-strain curve proposed by Popovics' (2) for the stress ascending portion was used in this investigation. This equa -tion (1) is shown in Table 3. Among the proposed equations， Popovics' equation is considered to be the most applicable to the stress-strain curves of various concretes Hatanaka et al.(3) proposed the equation to express the d巴scendingportions of the stress-strain curves of various concretes. This equation (2) is shown in Table 3. Equation (2) is considered to be the most applicable to the normalized stress-strain curves (吾-E curves，苔=σ/lio，旬。:compressive W/Cω% 0.5 _PR4 乙0 3.0

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(Nd 1 Fig. 5 B-Nd relation (m=0.8) strength， E =ε/ε0，ε。:strain at the compressive strength) of the various concretes. One example of the 5-E curves is shown in Fig.4. In this investigation， Equation (2) was used for expressing the stress-strain curve of concrete in the stress descending portion by setting the constsnts of B， Nd， and m according to the experimental data， respectively.

Fig.5 shows the relationship between B and Nd for the concrete of W /C = 60%， where the value of m is set to 0.8. It can been seen from Fig.5 that size of specimen and size of aggregate are considerably influenced by the relationship between Nd and B. Fig.5 shows出atthere are deviations from Nd-B r巴lationin the range of 5/φ< 3.0. In case of the

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Relationships of Reinforced Concrete Beams 18) W/C印 % 1.5 30mm 1 .日目。5 PR4 乙自 3.0 4.0 iNd) Fig. 6 Calibrated B司Ndrelation (mェ0.8) (8) W/Cω% ︿﹀

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INd1 Fig町 7 B-Nd relation calculat巴dby Eq (3) and (4) (solid and dotted line calc.) use of model concerte mad巴 with S/φ< 3.0 in 巴xperiment，points to which special attention should be paid are increase in deviation of stress strain curves.

Fig.6 shows the relationship betw巴巴nB and Nd

for the concrete of W/C=60% being adjusted by varing the parameter (size of specimen and size of aggregate)

The authors carri巴dout simple b巴ambending

test varying the dimensions made with the same mix proportion as this test program and examined experimentally the moment-curvature relationship The巴quation(3) and (4)shown in table 2 w巴re

obtained， where empirical constant B and Nd are expressed by using size of specimen S (cm) and size of aggregate (cm)， to compare the moment-curvature relationships obtained by the beam 100 1Cf"J 由 0.8 ~ 0.6 比」 a: 1-

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10→] Fig. 9 Stress-strain curves calculated by Fig.8， Eq.(1)and Eq. (2) (dotted lin巴:calc目) experiment and analitical results using the proposed stress-strain relations in equation(1)， (2) and (3)， (4). Fig.7 shows the B-Nd relations， where solid and dotted lines are calculated by巴quation(3) and

(4). Equation (3) and (4) can be fitt巴dwell enough to

the empirical B-Nd relations in Fig.5 Fig.8 shows the relationship between the varia -tion of relative stress S and relative strain E calcu -lated by the constant B and Nd in Fig.6 and equa -tion(1)， (2). Fig.9 shows the fitness of the proposed equation for the experimental stress-strain curves for the concret巴ofW/C=60%

5. PREDICTION OF MOMENT圃CURVATURE

RELATIONSHIPS OF MODEL RC BEAMS Th巴moment-curvaturerelationships obtained

from the experiment and from the analysis with the proposed stress-strain relations in equation (3) and (4) were compared， to discuss the applicability of the proposed stress-strain relations for the plastic rotational behavior of the critical s巴ctionof model

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224 Sachio KOIKE， Kenji OHISHI， Kazuo OKUFUJI and Nobuyuki OKUYA reinforced concret巴beams. The siz巴sof beam section tested (b x h， cm) were 4.46 x 8.92， 7.25 x 14.50， 9目68x 19.36， and 12 48 x 24.96 having its effective depth d=0.9h. Fig.10 shows outline of beam specimen. Stirrups w巴re placed closely to prevent sh巴ar failure. These

beams were made with th巴samemix proportions

as the pr巴vioustest program for the concrete of W /

Cニ60%. The mom巴nt-curvaturerelationships of

model beams subj巴ct巴dto two concentrated loads

acting at the third points (uniform moment span= 3h) were measured Fig.ll shows the moment.curvature relation

h

[

D

U

Fig. 10 Outline of beam specimen φ:25脇町bCfZ(WcmZ) 100 w/c:60 % pt :2_8~ φ:25臨 σ ( Wcm2) ships at critical section (curvature measurement length二 2h) of model beams (W/C=60%，φェ 25mm， Pt = 2.80%). Fig.12 shows the stress引ram curves calculated by equation (1)， (2) and (3)， (4)， and co即日teprism compressive strength and strain at the compressive str巴ngth obtained from the experiment for the concrete mad巴withφ=25mm Fig.13 shows the mom巴nt-curvaturer巴lationships at the critical section of the model beams (Pt = 2. 80%) calculated by thεstress-strain curves shown in Fig.12.Itis observed from the figure that the plastic rotational behavior of reinforeced concrete beams ar巴affectedby the size of beam specimens，

because of the size effect of the strength and stress strain relationships of concrete in the compressive zone of model beams. Fig.13 shows that close agreement between observed and calculated curva -ture is obtained except for the beam of b = 4.46cm

Fig.14 shows the analitically obtained moment-curvature curves calculated by Eq目 (1)，(2)

and (3)， (4) for the beams made with φ=20mm，

w/c:60 :;: φ:25 lIl!il W/C:ωz p.乙 8% 出IbcfZ(kg/Ql!2) 100

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1∞ 却 3∞ 4 a J 叩 φ.d(XIO-4) Fig. 11 Moment-curvature curves Fig. 12 Stress-strain curves calcu- Fig. 13 Moment-curvature curves

obtained from RC beam lated by Eq. (1) (2) and (3)， calculated by stress-strain bending test (4) curves shown in Fig.ll φ:20 mm w/c :ωz Wbd2 (kg/Ql!2 ) pt 乙8::;

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。

一一一一 10mm 一一一 15mm ----20mm -一一・ー25mm

一

30mm 1∞ 却 3∞ 4∞ 叩 φ'd(XI0-4) (c) b=15cm series Fig. 14 Analitically obtained moment-curvature curves calculated by Eq. (1) (2) and (3)， (4) (pt=2.80% ， σ~y = 4000kgf/cm2)

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Relationships of Reinforced Concrete Beams 30mm and b=15.0cm (Pt二 2.80%，(}sy=4000kgf/ cm'). Fig.14 (b) shows that the calculated curves of S/φ く3.0indicate too ductile comparing the calcu -lated with the experimental.Fig.14 (c) shows the calculated mom巴nt-curvatur巴curvesof reinforced concrete beam (b=15.0cm) made with the different size of aggregates. Because of the use of the con -crete of S/ゆ >3.0 in Fig.14 (c)， the phenomenon shown in Fig.14 (b) (bニ4.46-7.25cm)旦renot ob -served

The followings can be drawn on the plastic rotational capacity of reinforced concrete beam from the above and other numerical analyses. The plastic deformational capacity of reinforced con -cr巴tebeam is remarkably a妊ectedby the size of beam specimen and size of aggregate. The use of the concrete of S/φ< 3.0 is recommended to avoid

仏 CONCLUSION

Siz巴effecton inelastic deformational behavior

of concrete and reinforced concrete beam was discussed in this paper experimentally and analyti cally. The following results were obtained (1) Equation (2) proposed by Hatanaka et al' for the stress苧straincurves in the stress descending portion was used in this investigation (2) The compressive strength becomes larger and the descending portion of the str巴ss-strain curves becomes st巴巴perwith the increase in the size of concrete specimen. (3) The ductility of stress-strain curve in the stress descending portion decreas巴d with th巴 increase of the size of concrete specimen， compar苧 ing with the specimen series having the equal strength level. (4) Size of concrete specimen and size of aggre -gate markedly influence the relationship between empirical constant Nd and B in equation (2)， wh巴re the value of m is set to 0.8. These are shown in Fig 6. (5) The plastic rotational behaviors of rein -forced concrete beams are affected by the size of beam specimens， because of the size e妊ectof the concrete in the compressive zone of model beam

(6) The calculated moment-curvature cueve of reinforced concret巴beammade with the concrete of S/ゆく3.0indicate too ductile compared with the experiment (7) The plastic deformational capacity of rein -forced concrete beam is remarkably affected by the size of beam specimen and size of aggregate. The use of th巴 concrete of S/φ<3

。

目

forthe beam

experiment under白exureshould be avoided ACKNOWLEDGMENT

The authors are very grateful to Dr.Y oshio Kosaka (Nagoya Univ.)， Dr.Shigemitsu Hatanaka (Mie Univ，)目 Dr.Kazuo Yamada， Messers. Nao Kobayashi， Hiroyuki Kawabe， and Kouji Kosaka (Aichi Institute of Tech.) for their cooperation REFERENCES 1) Koike， S. : "Effect of Specimen size on Proba -bility Distribution of Concrete Strength， "CAJ Review of th巴GeneralMe巴ting，1981， pp.77-80 2) Kosaka， Y.， Morita， S. : "Reinforced Concrete Structur，巴"Maruzen， 1982， p.21 3) Hatanaka， S.， et al.:"Study on Stress-Strain Relashionship of Concrete in Plastic Hinge Range of Reinforced Concrete M巴mbers，"

Proc. of Annual Meeting of AIJ， Tokai Bran回

chi， 1986， pp.1l3-116.